We analyze the two remaining terms separately.
Let use define the notation \(\operatorname{tr}_{\le m}\)  as the partial trace over the first \(m\)  qubits, and \(\operatorname{tr}_{> m}\)  as the partial trace over the rest of the qubits.
Subadditivity of the von Neumann entropy \(S(\rho )\)  [1]}, [2]}, [3]} implies
\(S\left( \operatornamewithlimits{{\mathbb {E}}}_X \left(\operatorname{{\mathcal {E}}}_{X} \otimes \, \mathcal {I}\right)(\rho ^{t-1}_{\operatorname{{\mathcal {E}}}_X}) \right) &\le S\left( \operatorname{tr}_{\le m} \operatornamewithlimits{{\mathbb {E}}}_X \left(\operatorname{{\mathcal {E}}}_{X} \otimes \, \mathcal {I}\right)(\rho ^{t-1}_{\operatorname{{\mathcal {E}}}_X}) \right) + S\left( \operatorname{tr}_{> m} \operatornamewithlimits{{\mathbb {E}}}_X \left(\operatorname{{\mathcal {E}}}_{X} \otimes \, \mathcal {I}\right)(\rho ^{t-1}_{\operatorname{{\mathcal {E}}}_X}) \right),\\&\le S\left( \operatorname{tr}_{\le m} \operatornamewithlimits{{\mathbb {E}}}_X \left(\operatorname{{\mathcal {E}}}_{X} \otimes \, \mathcal {I}\right)(\rho ^{t-1}_{\operatorname{{\mathcal {E}}}_X}) \right) + m \log 2\\&= S\left( \operatorname{tr}_{\le n} \operatornamewithlimits{{\mathbb {E}}}_X \rho ^{t-1}_{\operatorname{{\mathcal {E}}}_X} \right) + m \log 2.\) 
